Infinitely many bubbling solutions and non-degeneracy results to fractional prescribed curvature problems

TL;DR

This paper constructs infinitely many multi-bubbling solutions for a fractional prescribed curvature problem, revealing new solution structures and establishing non-degeneracy results using advanced analytical techniques.

Contribution

It introduces a novel construction of multi-bubbling solutions concentrating on surfaces of an oblate cylinder for fractional curvature equations, and proves their non-degeneracy.

Findings

Existence of infinitely many bubbling solutions with specific concentration patterns.

Development of a new method to prove non-degeneracy of solutions.

Application of Lyapunov-Schmidt reduction and Pohozaev identities in fractional problems.

Abstract

We consider the following fractional prescribed curvature problem $$(-\Delta)^s u= K(y)u^{2^*_s-1},\ \ u>0,\ \ y\in \mathbb{R}^N,\qquad (0.1)$$ where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geqslant4$ and $2^*_s=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $K(y)$ has a local maximum point in $r\in(r_0-\delta,r_0+\delta)$. First, for any sufficient large $k$, we construct a $2k$ bubbling solution to (0.1) of some new type, which concentrate on an upper and lower surfaces of an oblate cylinder through the Lyapunov-Schmidt reduction method. Furthermore, a non-degeneracy result of the multi-bubbling solutions is proved by use of various Pohozaev identities, which is new in the study of the fractional problems.